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Theorem cnvin 5698
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvin (𝐴𝐵) = (𝐴𝐵)

Proof of Theorem cnvin
StepHypRef Expression
1 cnvdif 5697 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴(𝐴𝐵))
2 cnvdif 5697 . . . 4 (𝐴𝐵) = (𝐴𝐵)
32difeq2i 3868 . . 3 (𝐴(𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
41, 3eqtri 2782 . 2 (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐵))
5 dfin4 4010 . . 3 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
65cnveqi 5452 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
7 dfin4 4010 . 2 (𝐴𝐵) = (𝐴 ∖ (𝐴𝐵))
84, 6, 73eqtr4i 2792 1 (𝐴𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  cdif 3712  cin 3714  ccnv 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274
This theorem is referenced by:  rnin  5700  dminxp  5732  imainrect  5733  cnvcnv  5744  cnvcnvOLD  5745  pjdm  20273  ordtrest2  21230  ustexsym  22240  trust  22254  ordtcnvNEW  30296  ordtrest2NEW  30299  msrf  31767  elrn3  31980  pprodcnveq  32317
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